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Theorem ndmovcom 6023
Description: Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)
Hypothesis
Ref Expression
ndmov.1  |-  dom  F  =  ( S  X.  S )
Assertion
Ref Expression
ndmovcom  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  ( B F A ) )

Proof of Theorem ndmovcom
StepHypRef Expression
1 ndmov.1 . . 3  |-  dom  F  =  ( S  X.  S )
21ndmov 6020 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  (/) )
3 ancom 437 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  <->  ( B  e.  S  /\  A  e.  S )
)
41ndmov 6020 . . 3  |-  ( -.  ( B  e.  S  /\  A  e.  S
)  ->  ( B F A )  =  (/) )
53, 4sylnbi 297 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( B F A )  =  (/) )
62, 5eqtr4d 2331 1  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   (/)c0 3468    X. cxp 4703   dom cdm 4705  (class class class)co 5874
This theorem is referenced by:  addcompi  8534  mulcompi  8536  addcompq  8590  addcomnq  8591  mulcompq  8592  mulcomnq  8593  addcompr  8661  mulcompr  8663  addcomsr  8725  mulcomsr  8727  addcomgi  27764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715  df-iota 5235  df-fv 5279  df-ov 5877
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